球码：豪斯多夫维数与填充维数的编码刻画
Ball codes: A coding characterization of Hausdorff and packing dimensions
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中文总结 AI 辅助
该研究提出球码概念，通过编码理论刻画\boldsymbol{\text{R}}^n中的豪斯多夫维数与填充维数，还推导了J. Lutz和N. Lutz的点集原理。
中文摘要 AI 辅助
我们证明了\boldsymbol{\text{R}}^n中豪斯多夫维数与填充维数的纯经典编码理论刻画。一个\textbf{球码}为\boldsymbol{\text{R}}^n中的闭球分配名称：它是从有限二进制串的无前缀集到球的部分映射。对任意非空子集\boldsymbol{E}\boldsymbol{\text{R}}^n，\boldsymbol{E}的豪斯多夫维数是所有球码中，\boldsymbol{E}中每个点\boldsymbol{x}被包含它的球的名称描述时的下渐近率的上确界的最小值；填充维数则由上渐近率按相同方式得到。该豪斯多夫维数刻画是组合源的Ryabko编码定理的欧几里得对应。我们通过上修正盒计数维数对填充维数的标准刻画得到填充维数刻画，无需将\boldsymbol{\text{R}}^n编码为序列空间。随后，我们通过将最小化球码替换为有理球码，并将所得可数码本存储于谕示中，推导出J. Lutz与N. Lutz的点集原理。
英文摘要
We prove a purely classical, coding-theoretic characterization of Hausdorff and packing dimensions in \(\mathbb R^n\). A \emph{ball code} assigns names to closed balls of \(\mathbb R^n\): it is a partial map from a prefix-free set of finite binary strings to balls. For every nonempty \(E\subseteq\mathbb R^n\), the Hausdorff dimension of \(E\) is the minimum, over all ball codes, of the supremum over \(x\in E\) of the lower asymptotic rate at which \(x\) can be described by names of balls containing it; the packing dimension is obtained in the same way from the upper rates. The Hausdorff characterization is a Euclidean counterpart of Ryabko's coding theorem for combinatorial sources. We obtain the packing characterization from the standard characterization of packing dimension by upper modified box-counting dimension, without encoding \(\mathbb R^n\) into a sequence space. We then derive the point-to-set principle of J.~Lutz and N.~Lutz by replacing a minimizing ball code with a rational one and storing the resulting countable codebook in an oracle.
发表机构
- Meiji University（明治大学）
机构由 AI 辅助整理，请以论文原文为准。
